Centripetal force: Moment of force = Force x Perpendicular distance from fulcrum.
If a force is applied upward, the force will have an anticlockwise moment. If a force is applied downward, it will have a clockwise moment. The resultant force applied is found by:
Resultant Moment = Total clockwise moment - Total anticlockwise moment.
A beam supporting a load is subject to a bending moment. A beam or joist is usually laid with its narrower face toward the load, so that its width bears the load. The load is borne perpendicular to its length, and the beam needs to be laid for maximum stiffness. The stiffness (elastic modulus) of a material is its resistance to elastic deformation:
Stiffness = Load / Deflection
The weakest point on a beam is in the centre of its length. The load on the beam is counter-balanced by internal forces in the beam and the upthrust from the supporting walls at each end of it. The greater the span of the beam, the greater the effect of a load, and the greater the internal forces built up in the beam as these resist bending moment created by the load. The magnitude of the bending moment depends on the amount of load and distance from pivot (supporting wall).
Stress = Force / Area
The distance of the load from the pivot is the moment arm. So Moment = Load x Moment arm.
When a beam is under load, its top is compressed while its underside is in tension, as an upward curve is created.
Strain = L2 - L1 / L1
where L1 is the original length and L2 is the final length
or Strain = Change in length / Original length x 100 to express as a percentage.
These elastic strains must be counter-balanced by internal stress. The stresses are related to strain by Young's Modulus:
Stress = Young's Modulus x Strain.
Rearranging gives: Young's Modulus = Stress / Strain
So the External Moment = Stress x Area x Moment arm perpendicular to the force, is balanced by Internal Moment = Stress x Area x Moment arm parallel with the force. Tensile stress gives rise to tensile forces while compressive stress gives rise to compressive forces. Tensile force is denoted by a positive value and compressive force is given a negative value in calculations.
---------------------><-------------------- Moment Arms
Liquid Pressure:Pressure is exerted at all points in a liquid due to its weight. The greater the depth of a liquid, the greater the pressure exerted at a given depth. Pressure also increases with density. The pressure at any given point acts equally in all directions.
Because liquids cannot be compressed much, their volume can't be decreased with pressure, and so they transmit pressure applied. Examples of common uses include hydraulic jacks and car brakes.
Pressure = Density x Gravity x Height (of liquid)
and Total Pressure = Dgh + Atmospheric Pressure
To calculate transmitted pressure, use P = F / A, or rearranging to find an applied force gives F = PA.
Gas pressure is caused by molecular kinetic energy being exerted on a container's walls by the gas. The smaller the volume, the greater the gas pressure.
Gas Pressure = Atmospheric pressure + Excess pressure of gas above atmosphere.
Air exerts atmospheric pressure. Normal atmospheric pressure is 101,325 Pa, and is equal to 1 atm. It acts from all directions on the earth. The body uses blood pressure to counter-balance the force of atmospheric pressure. The higher an object is, the less the atmospheric pressure. Aircraft are seal pressurised to about equal to atmospheric pressure at 1500 feet. If decompression occured, objects and the air would be sucked out. Barometers measure the air pressure pushing downward . 1 atm is equal to 760 mm Hg on a mercury barometer.
On the earth's surface exist regions of high and low pressure. A high pressure region is an anticyclone, and a low pressure region is a cyclone or depression. Isobars on a weather map join up regions of equal pressure. Winds tend to blow into low pressure regions, and so the weather in these is usually unsettled. Similarly, when isobars are close together, the wind is strong, blowing into areas where isobars are further apart.
Atmospheric pressure on weather charts are usually given as bars, with 1 atm equalling 1 bar.
Power is the amount of work done in 1 second:
Power = Work / Time, or Energy transfer / Time
This is measured in J/s or watts, W.
Energy is changed from one form to another by transducers. However, it can not be made or destroyed. Examples of energy conversion: Burning of fossil and vegetable fuels to convert chemical into thermal energy; nuclear fuels convert some of their mass into thermal energy; wind and wave energy is used as kinetic energy to drive turbines and generators to give electrical energy; the sun converts mass into electro-magnetic energy giving heat and light and in turn wind and wave energy; a piezoelectric lighter converts elastic potential energy into electrical energy; in the earth radioactive decay generates geothermal energy; in plants, chloroplasts convert electro-magnetic energy from light into chemical energy.
Kinetic energy is the energy of motion. Potential energy is the energy stored by a body because of its positon. Gravitational potential energy increases with the height of an object. A charge in an electrical field has electric potential energy, and for a magnet in a magnetic field, it is magnetic.
The body converts chemical energy into thermal, kinetic and new chemical enrgy. It also may convert some of its mass into energy.
A coal power station converts chemical energy from fuel into thermal and potential energy, then these in turn are converted into kinetic energy, and finally electrical energy. This is done by means of the boiler - turbine - generator system.
Nuclear power stations replace chemical with nuclear energy, & the boiler with the reactor.
Regardless of the system type, there is always energy loss. Work output is less than work input.
Efficiency = Work output / Work input or Energy output / Energy input
multiply the result by 100 to obtain a percentage.
If force is given off in a field then another force may work with or against it, e.g. A negative charge in a positive field will be attracted by it. The field does work to attract it, and the force's potential energy falls. If a positive force is in a positive field, it will be repelled. If the force is pushed into the field, work is done against the field, and the potential energy of the force will increase.
The kinetic energy of a body is given by:
K.E. = 1/2 mv2
If an object falls from a height, its potential energy is converted into kinetic energy:
Change in P.E. = Weight x Change in height, or Wh.
Mass is also a form of energy. The relationship between mass and its energy equivalence is given by Eintstein's Theory of Relativity equation:
Energy = Mass x Electromagnetic radiation in a vacuum2
or E = mc2
All matter is made up of particles, whether atoms or molecules of atoms. In solids, the atoms are tightly packed in a crystal latice structure. The atoms vibrate about their rest positions. As they approach other atoms, they are repelled in opposite directions. As they approach rest position, they are attracted toward eachother again. The attractive force in solids is stronger than in liquids or gases. Solids cannot expand or compress much. They are subject to tensile and compressive force, but to a limited extent. Solids have a greater density than liquids or gases.In liquids, the atoms have no definite arrangement and only slight attractive force between atoms. Liquids are able to flow. They can be compressed a little more than solids and are less dense. This is due to their atomic masses. Mercury has a high density, and is the only liquid metal at room temperature.
In gases, there is no pattern and the atoms move independently of eachother. They move in straight lines until repelled by other atoms or obstacles. The attractive force only acts when they are close enough to another atom. This is the kinetic theory of gases. A gas will always exert pressure on its container, as the atoms hit the inside of it. Gases can be compressed a great deal more than solids or liquids, and are less dense than either.
Brownian motion occurs in fluids due to atomic motion and collision. Particles of a substance appear to move about at random. Small molecules can move larger particles despite their greater inertia. As the small molecules collide with bigger ones, they decelerate rapidly, exerting a large force on them during the collision. The larger particles accelerate reaching a velocity.
Diffusions occur as fluids mix. Smells diffuse as particles of a substance travel through air. The lighter the atoms or molecules of a substance, the faster the rate of diffusion.
Molecules may travel at different speeds and have different kinetic energies. In liquids, this causes evaporation. When atoms near the surface have enough kinetic energy to overcome attractive forces in the liquid, they escape, filling the air above with vapour. As a result, the average kinetic energy and temperature of the liquid drops. Evaporation always cools. Because of the state change to vapour, the atoms will increase their volume. The difference between a vapour and a gas is that a vapour can be turned back into a liquid by compression alone, whereas a gas needs to be cooled and compressed to be turned into a liquid.
3 factors effecting a rate of evaporation are temperature, surface area, and presence of any draughts. If these factors are increased, evaporation will increase also.
If thermal energy is applied, the temperature rises, and the atomic motion increases. In solids, the atoms will vibrate over greater distances in the same amount of time. In liquids, the atoms will move about more rapidly and collide more often. In gases, the speed and kinetic energy of the atoms increases, and hence, the pressure.
Absolute zero: If cooling is applied to a gas, its pressure is reduced. If cooling continued, eventually gas pressure would be reduced to zero. This is absolute zero, or 0 Kelvin. This is only a hypothesis based on an ideal gas.
Boyle's Law applies to a constant mass of gas at constant temperature, but variable pressure and volume:
Pressure x Volume = Constant
or to indicate a change of inverse proportion: p1 V1 = p2 V2
i.e. If pressure is doubled, volume is halved. If pressure is trippled, volume is reduced to 1/3. A decrease in volume will also cause an increase in density.
The Pressure Law applies to a constant mass of gas at constant volume, but variable temperature and pressure:
Pressure / Temperature = Constant
The relationship is one of direct proportion. If the temperature is doubled, the pressure also doubles. To indicate a change:
p1 / T1 = p2 / T2
If water is cooled, its volume will decrease until the temperature reaches about 277K. As it continues to cool, the volume begins to increase until 273K, when it changes to ice and there is a large increase in volume.
For a fixed mass of an ideal gas at constant pressure:
Volume / Temperature (K) = Constant
or to indicate a change of direct proportion:
V1 / T1 = V2 / T2
The General Gas Law:
A gas may start off or finish at 273K and 1 atmosphere. This is standard temperature and pressure, stp, for a gas.
pressure x Volume / Temperature (K)
and to indicate a change:
p1V1 / T1 = p2V2 / T2
This is an example of equivalent statements. e.g. A fixed mass of gas at 300K and 1 atm has a volume of 200cm3. To calculate its volume if he temperature rises to 400K and pressure to 1.5 atm:
1 x 200 / 300 = 1.5 x V2 / 400. V2 = 178 cm3.
Temperature is a measure of the kinetic energy in a substance. The greater the temperature, the faster the kinetic energy of the atoms. If 2 bodies at different temperatures are placed next to eachother, heat will flow from the warmer to the cooler body. The internal kinetic energy of the cooler body will increase and its temperature will rise:
Heat lost by hot body = Heat gained by cooler body
To convert Celsius to Kelvin:
Celsius + 273 = Kelvin
To convert Kelvin to Celsius:
Kelvin - 273 = Celsius
Calibrating a mercury thermometer:
The lower fixed point on the Celsius scale is the temperature of pure melting ice, or 0o Celsius. The upper fixed point is the temperature of steam above boiling water at normal atm. If these 2 fixed points are marked on a mercury glass thermometer, and the distance between these was 150mm, each degree must be calibrated 150/100 mm apart. A narrower capillary tube would increase the sensitivity and each degree could be placed further apart.
Thermocouples consist of 2 wires made of 2 different metals. When attached to a galvanometer, usually calibrated in Celsius, the temperature obtained at the hot junction of the 2 wires is recorded. Thermocouples are designed for industrial high temperature environments. They range from about -40 - 125o C up to about 1100 - 1600o C.
Thermistors are electronic thermometers. They are made from a semi - conductor whose electrical resistance is reduced as temperature increases. The passing current gives the temperature reading. Thermistors are also used to prevent overloading in a circuit.In science, the absolute or Kelvin scale is used. The fixed point of the Kelvin scale is the triple point, or 273.16K. The melting point of pure ice is 273K. 0K, or absolute zero = -273o C. For calculations dealing with gases, change temperatures into K.
Energy applied to a body is related to the resultant temperature rise by:
Energy supplied = Thermal capacity x Temperature rise
Thermal capacity = Energy supplied / Temperature rise
in J/oC or J/K.
Why some hot objects burn: A hot object cools down and releases heat. Some objects will burn, while others don't. The objects that burn release greater amounts of heat as they cool. These objects have a higher thermal capacity.
Each substance has its own specific heat capacity:
Specific heat capacity = Energy supplied / Mass x Temperature rise
For energy supplied by a heater: Energy supplied = Wattage x Time
Energy supplied = Mass x Specific heat capacity x Temperature rise
in J/kg oC or J/kg K.
As energy is supplied, it produces heat which increases the internal kinetic energy of the substance, and hence, its temperature. If energy is removed, the temperature will fall as heat is given out during cooling.
If the same amount of heat is applied to the same mass of 2 different substances, the larger temperature rise will occur in the one with the lowest specific heat capacity. If 2 bodies at 2 different temperatures are joined, the final temperature Tf of both is found by:
Temperature fall of hot body is T1 - Tf
Temperature rise of cold body is Tf - T2
m1c1(T1 - Tf) = m2c2(Tf - T2)
Tf = m1c1T1 + m2c2T2 /(m2c2 + m1c1)
(m = mass, c = specific heat capacity)
For equal masses of the same substance at different temperatures:
Tf = T1 + T2 / 2
The final temperature is the average of the 2.
Melting and boiling points:
If ice is heated from below 0oC, its temperature will rise until it reaches 0oC. When this happens, it will remain at 0oC and will start to melt. When it has all melted into liquid the temperature will again start to rise. When it reaches 100oC the temperature will stop rising and the water will begin to boil. It will turn into steam, at which time there will be a large increase in its volume. When it has all turned into vapour, the temperature will start to increase again. The vapour is now a gas. Melting and boiling are 2 points at which there is no rise in temperature from continued heat input.
If a substance is impure, the melting point will be lowered. Impurities also increase the boiling point.
Condensation and solidification are the reverse processes of boiling and melting. If a gas is cooled, it will condense into a liquid. If cooled further, the liquid will eventually solify. Impurities lower the freezing point.
Latent heat is the heat used to melt or boil a substance, and doesn't cause a rise in temperature. Latent heat of fusion is applied to melt a substance. To vapourize the substance, latent heat of vapourisation is applied. Latent heat enables atoms to break free of the lattice structure of solids and increases their potential energy. During vapourisation, the latent heat frees the atoms further such that they are able to move independently of eachother. Vapour works against the atmosphere as it pushes outward, so the higher the atmospheric pressure, the higher the boiling point.
Melting or boiling are changes of phase. Latent heat increases atomic potential energy, without increasing the temperature of the substance. Heat normally increases the kinetic energy of the atoms, and the temperature. Large amounts of latent heat are required to melt or boil substances. Melting points and boiling points can be found from a temperature - time graph.
Normally heat flows from the warmer to the cooler substance. Refrigerators make heat flow in the opposite direction by setting up a cold convexion current.
During boiling or evaporation molecules escape from the liquid. However, while evaporation occurs at all temperatures, boiling only occurs at the boiling point. Evaporation involves a loss of surface molecules and results in cooling. Boiling involves a loss of molecules from the bulk of a substance, and does not result in cooling as heat is being applied. Pressure exerted by a vapour equals atmospheric pressure.
Each substance has its own specific latent heat of fusion and vapourisation. The slh of fusion is the amount of heat needed to change 1kg of the solid into liquid:
Heat required = Mass x Specific latent heat of fusion
or W = mL
e.g. The amount of heat required to melt 10g of ice if L = 3.4 x 105J/kg:
m = 10g = 10-2kg, L = 3.4 x 105J/kg
W = 10 -2 x 3.4 x 105J
Similarly, the amount of heat needed to change 1 kg of substance into vapour is the specific latent heat of vapourisation:
Heat required = mass x specific latent heat of vapourisation
using the same formula.
e.g. Amount of heat needed to change 0.5kg of water to vapour if L = 2.3 x 106J/kg:
m = 0.5kg, L = 2.3 x 106J/kg
W = 0.5 x 2.3 x 106 = 1.15 x 106J
= 1.15 MJ
The high specific latent heat of vapourisation of water means that steam will burn on contact more readily than boiling water at 100oC.
e.g. To change ice at 0oC into vapour at 100oC, the heat required is given by:
W = mLf + mc x temperature change x mlv
m = mass of ice or water, c = specific heat capacity of water, Lf = specific latent heat of fusion, Lv = specific latent heat of vapourisation.
Determination of specific latent heat of fusion of ice in the laboratory:
An immersion heater is placed in a funnel of ice, on top of a beaker of known mass. The heater and timer is switched on. After 5 minutes the heater and clock are stopped and the beaker weighed. The new mass is noted, and the mass of the beaker subtracted. To obtain the specific latent heat of fusion:
W = mL and W = Pt
(P = wattage of heater, t = time in seconds).
To obtain the specific latent heat of vapourisation of water, a beaker of boiling water is weighed and the mass noted. Heat is reapplied and the clock started. After a suitable amount of time, the clock and heater are stopped, and the new mass noted. The difference between the masses gives the mass of water vapourized:
W = mL and W = Pt as before.
Methods of heat transfer: Conduction, convection and radiation
Conduction occurs in solids, liquids and gases. Heat moves through the substance, but the substance itself does not move. Metals are the best conductors. Metals feel colder to the touch than other solids, as these conduct heat away from the source faster. Tests can be used to find the better conductor of a group of substances. Heat will travel faster in it. The following substances are listed in order of best to worst heat conductivity: copper, aluminium, iron, brick, glass, water, polythene, air, plaster, expanded polystyrene.
The heat conductivity rate in a substance depends on type of substance, thickness of sample, temperature difference (temperature at hot face - temperature at cold face)across the thickness, size of hottest area. Heat will flow from hot area to cold area, and the amount of heat conducted every second is the conductivity rate. In solids such as metals, heat is conducted by electrons moving through them. The atoms in the crystal lattice also vibrate over greater distances when heated.
Convection is the movement of heat through a substance by movement of the substance itself. Convection can only occur in liquids or gases. As parts of a substance are heated, they expand and become less dense. The denser surroundings push these lighter expanded parts to the top. Heat appears to rise. This fluid motion is convection currents, caused by differences in density. Thermals in air are cunvection currents.
Radiation is heat transfer via infra-red rays. Radiation can travel in a vacuum and doesn't require a substance. Radiation generates both heat and current, and lowers electrical resistance. Dull dark surfaces are good emitters and absorbers of radiation. Bright light surfaces reflect rather than absorb radiation, and won't become as hot from it as a darker surface.
Insulators inhibit the flow of heat. A thermos flask is designed to inhibit all 3 types of heat flow, with a vacuum sealed hollow walled glass container that is silvered. Slight conduction does occur up the walls. Buildings with cavity walls and doubled glazed windows cut convection and conduction, as well as noise. About 15% of heat loss is through the windows. Fibre - glass insulation bats in the roof also cut conduction through the roof, the prime source of heat loss in a house. Plastic foam or fibre - glass in the walls will reduce convection.
Outer wall (brick/ cement block)||__|cav.|_||Inner wall with insulation
Waves and wave motion:
Waves transfer energy. If water is disturbed, the resulting waves move up and down. Anything on its surface will also move up and down with the waves. All the troughs and crests move along at the same rate, but each bit of the surface is slightly out of step with the next. The wave transfers the energy from its source to other objects on the surface. As the source and other objects move up and down, their potential energy changes. The average position of the surface is its normal rest position. If the wave is moving along in 1 direction, it is a transverse wave. In a transverse wave, the substance being disturbed moves along at right angles to the disturbance or propogation of the wave:
| /\/\/\/\/\/\/\/\/\/\/\=> Wave moves along transferring energy
Other examples of transverse waves are light or a rope shaken up and down.
The speed of a wave is the distance it travels in 1 second, v. The shortest distance between 2 in-phase points, such as the tops of 2 crests is the wavelength, L (lambda). The number of complete wavelengths passing a given point is the wave's frequency, in Hz, f. The distance from the trough to the rest position is the amplitude of the wave, or maximum displacement from rest position. The time it takes for 1 wave to pass is its period, T.Each time the disturbing object vibrates, the wave generated moves along by 1 wavelength. If the object makes f vibrations per second, then its freuqency is f Hz, and the distance moved by the wave will fL. Because this distance is moved in 1 second, it is the wavespeed, v:
Waves that travel in straight lines such as described above, are plane waves. If a ball was used to propagate a wave the resultant waves would spread out from it and would be circular. Wavefronts would be a series of concentric circles.If incident plane waves are reflected off a plane boundary, the angle between the incident waves and the normal (imaginary line at 90o to the plane boundary) is the angle of incidence. The angle between the normal and the direction of the reflected waves is the angle of reflection:
If incident plane waves travel from a deeper region into a shallower region, they will be refracted and their direction changes in the shallow region. Their wavelength will also decrease in the shallow region. The angle between the normal and the direction of travel of the refracted waves is the angle of refraction. The angle of incidence doesn't = the angle of refraction.
If 2 balls are used to generate circular waves, the waves start off in phase. Some of the waves will have travelled 1, 2, 3, etc. wavelengths, and these waves will combine in phase and make a wave of double the amplitude. This is constructive interference and occurs when troughs coincide with troughs. In other directions, the waves will travel 1.5, 2.5, 3.5, etc wavelengths. These waves will will cancel eachother out. This is destructive interference This occurs when troughs coincide with crests.
If plane waves are incident on a slit barrier with only a small slit, the waves are diffracted to either side at an angle. If the slit is too wide, no diffraction occurs. If incident plane waves strike a concave barrier, they will be reflected as circular waves. They will meet at the focus point of the concave barrier. If plane waves are incident on a convex barrier, the reflected circular waves will appear to diverge from a focus point behind the the convex barrier. But they are not actually there. If circular waves are incident on a plane boundary the boundary will reverse the curvature of the reflected waves. They will appear to have come from a point equidistant behind the plane barrier as the propagator of the wave is in front.
Refraction may also be caused by a change in speed. As the incident plane wave enters the shallower region, the speed as well as the wavelength of the refracted wave is slower because: wave speed = frequency x wavelength. It is this slowing of the waves that causes the change in direction. To calculate the refractive index:
Speed over deep area (faster)/ Speed over shallow area (slower)
If a spring is pulled and released horizontally, a wave will travel along the spring. This wave consists of rarefractions, or stretched sections, and compressed sections, or compressions. These travel along the spring.
|||| | | | ||||| | | | ||||| | | | ||||| => wave travel
The distance from the middle of 1 compression to the middle of the next is 1 wavelength. In a longitudinal wave the direction of travel of the wave is the same direction as that of the disturbing force. The coils move horizontally as does the wave motion. Sound is a longitudinal wave.